01) Differentiation: Fundamentals

Differentiation is a method to find the "Rate of Change" or "Change Per Unit Time" of an equation, this can mean the change in distance on a journey per hour, the change in volume of water in a leaking tub per minute, or even the change in distance of an accelerating race car per second. These results can be expanded further: finding the velocity of an accelerating ball at a certain distance, finding the current change in volume of a tub of water at a specific height, the finding the maximum area of an object with a constant perimeter, and more. As long as there is a change in a measurement that correlates to the change of another measurement, differentiation can be of use.

First and foremost, the differentiated result of a function is known as a derivative. For example, speed can be considered the "Change of Distance Per Unit Time", so speed is the 'Derivative of Distance".

Second of all, the amount of times something can be differentiated is infinite (unless there has yet to be a found derivative or method to the strange function you're attempting, which is unlikely but not impossible)! The way this is represented is simple, the differentiation of function "f(x)" is the "Derivative of f(x)", and the differentiation of the "Derivative of f(x)" is the "Derivative of the Derivative of f(x)" also known as the "Second Derivative of f(x)".

An example would be acceleration, acceleration can be considered the "Change of Speed Per Unit Time", thus acceleration is the "Derivative of Speed", as well as the "Derivative of the Derivative of Distance" also known as the "Second Derivative of Distance".

Thirdly, there are many mathematical representations of differentiation.

    i) d/dx

    Personally, this is an absolutely fundamental denotation of differentiation, as most shorthand of derivatives assume that there are only two variables in the equation and can only show a constant amount of derivatives, this denotation is able to specify which variable the derivative is of. A simple example of this would be a "Differentiation by x" for y that increases for every 3 units of x^5.

Further differentiation looks like this:

More-than-2 variable equation will be shown in a later chapter.

    ii) y'

    A shorthand for 2 Variable Equation. Assuming the two variables are x and y, then the derivative of y can be represented as y'. Differentiating further will increase the amount of apostrophes, a double derivative of y would be represented as y'', a triple derivative is y''', and so on.

The differentiation of a function, in this case represented as f(x), is known as a derivative, which in this case is represented as f'(x).

And every further differentiations will continue to add more apostrophes.

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